On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields

Phitthayathon Phetnun, Narakorn Rompurk Kanasri and Patiwat Singthongla

International Journal of Mathematics and Mathematical Sciences, 2021, vol. 2021, issue 1

Abstract: For a Gaussian prime π and a nonzero Gaussian integer β = a + bi ∈ ℤ[i] with a ≥ 1 and β≥2+2, it was proved that if π = αnβn + αn−1βn−1 + ⋯+α1β + α0≕f(β) where n ≥ 1, αn ∈ ℤ[i]\{0}, α0, …, αn−1 belong to a complete residue system modulo β, and the digits αn−1 and αn satisfy certain restrictions, then the polynomial f(x) is irreducible in ℤ[i][x]. For any quadratic field K≔ℚm, it is well known that there are explicit representations for a complete residue system in K, but those of the case m ≡ 1 (mod4) are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

Date: 2021
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https://doi.org/10.1155/2021/5564589

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jijmms:v:2021:y:2021:i:1:n:5564589

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